Lines of full rank matrices in large subspaces

TL;DR

This paper extends classical results on large subspaces of matrices by showing that under certain codimension constraints, lines directed by low-rank matrices intersect the subspace in matrices of full rank.

Contribution

It proves a new geometric property of large subspaces of matrices, relating codimension bounds to the existence of lines with full-rank matrices intersecting the subspace.

Findings

If codim S < n-1, then lines directed by matrices of rank less than p intersect S in full-rank matrices.

The result generalizes Flanders' theorem by involving lines and directions in matrix spaces.

Provides conditions under which low-rank matrices can be connected to full-rank matrices within large subspaces.

Abstract

Let $n$ and $p$ be non-negative integers with $n \geq p$, and $S$ be a linear subspace of the space of all $n$ by $p$ matrices with entries in a field $\mathbb{K}$. A classical theorem of Flanders states that $S$ contains a matrix with rank $p$ whenever $\mathrm{codim} S <n$. In this article, we prove the following related result: if $\mathrm{codim} S<n-1$, then, for any non-zero $n$ by $p$ matrix $N$ with rank less than $p$, there exists a line that is directed by $N$, has a common point with $S$ and contains only rank $p$ matrices.